Question

can someone help me with the following problems :(

Problem 1

Design a digital circuit that determines the amount of 1’s and 0’s in an 8-bit binary number and check whether both quantities are odd or even.

Problem 2

Create a medium subtractor circuit, which allows subtracting the x-y bits which generate the difference in the output and the loan in the output p.

Answer 1

ANSWER OF 1;

Write down the truth table. The output is a 1 if the combination of inputs is odd, otherwise 0.

1. A B C ODD
2. 0 0 0 0
3. 0 0 1 1
4. 0 1 0 1
5. 0 1 1 0
6. 1 0 0 1
7. 1 0 1 0
8. 1 1 0 0
9. 1 1 1 1

Now at this point you could do a formal analysis and develop a minimised logic equation, but from the examination of the truth table, you can immediately observe the following:

• The first four outputs are the inverse of the second four, and that corresponds to the state of A
• The first four outputs are the XOR of B and C
• The second four outputs are XNOR of B and C

The XOR gate can be thought of as a controlled inverter - if one of its inputs is 0, the output follows the other input, if it’s 1, the output is the inverse of the input. So to implement this we just need two XOR gates:

ANSWER 2;

As their name implies, a Binary Subtractor is a decision making circuit that subtracts two binary numbers from each other, for example, X – Y to find the resulting difference between the two numbers.

Unlike the Binary Adder which produces a SUM and a CARRY bit when two binary numbers are added together, the binary subtractor produces a DIFFERENCE, D by using a BORROW bit, B from the previous column. Then obviously, the operation of subtraction is the opposite to that of addition.

We learnt from our maths lessons at school that the minus sign, “–” is used for a subtraction calculation, and when one number is subtracted from another, a borrow is required if the subtrahend is greater than the minuend. Consider the simple subtraction of the two denary (base 10) numbers below.

123	X	(Minuend)
– 78	Y	(Subtrahend)
45	DIFFERENCE

We can not directly subtract 8 from 3 in the first column as 8 is greater than 3, so we have to borrow a 10, the base number, from the next column and add it to the minuend to produce 13 minus 8. This “borrowed” 10 is then return back to the subtrahend of the next column once the difference is found. Simple school math’s, borrow a 10 if needed, find the difference and return the borrow.

The subtraction of one binary number from another is exactly the same idea as that for subtracting two decimal numbers but as the binary number system is a Base-2 numbering system which uses “0” and “1” as its two independent digits, large binary numbers which are to be subtracted from each other are therefore represented in terms of “0’s” and “1’s”.

Binary Subtraction

Binary Subtraction can take many forms but the rules for subtraction are the same whichever process you use. As binary notation only has two digits, subtracting a “0” from a “0” or a “1” leaves the result unchanged as 0-0 = 0 and 1-0 = 1. Subtracting a “1” from a “1” results in a “0”, but subtracting a “1” from a “0” requires a borrow. In other words 0 – 1 requires a borrow.

Binary Subtraction of Two Bits

0	1	1	(borrow)1→ 0
– 0	– 0	– 1	– 1
0	1	0	1

For the simple 1-bit subtraction problem above, if the borrow bit is ignored the result of their binary subtraction resembles that of an Exclusive-OR Gate. To prevent any confusion in this tutorial between a binary subtractor input labelled, B and the resulting borrow bit output from the binary subtractor also being labelled, B, we will label the two input bits as X for the minuend and Y for the subtrahend. Then the resulting truth table is the difference between the two input bits of a single binary subtractor is given as:

2-input Exclusive-OR Gate

Symbol	Truth Table
2-input Ex-OR Gate	Y	X	Q
	0	0	0
	0	1	1
	1	0	1
	1	1	0

As with the Binary Adder, the difference between the two digits is only a “1” when these two inputs are not equal as given by the Ex-OR expression. However, we need an additional output to produce the borrow bit when input X = 0 and Y = 1. Unfortunately there are no standard logic gates that will produce an output for this particular combination of X and Y inputs.

But we know that an AND Gate produces an output “1” when both of its inputs X and Y are “1” (HIGH) so if we use an inverter or NOT Gate to complement the input X before it is fed to the AND gate, we can produce the required borrow output when X = 0 and Y = 1 as shown below.

Then by combining the Exclusive-OR gate with the NOT-AND combination results in a simple digital binary subtractor circuit known commonly as the Half Subtractor as shown.

A Half Subtractor Circuit

A half subtractor is a logical circuit that performs a subtraction operation on two binary digits. The half subtractor produces a sum and a borrow bit for the next stage.

Half Subtractor with Borrow-out

Symbol	Truth Table
	Y	X	DIFFERENCE	BORROW
	0	0	0	0
	0	1	1	0
	1	0	1	1
	1	1	0	0

From the truth table of the half subtractor we can see that the DIFFERENCE (D) output is the result of the Exclusive-OR gate and the Borrow-out (Bout) is the result of the NOT-AND combination. Then the Boolean expression for a half subtractor is as follows.

For the DIFFERENCE bit:

D = X XOR Y = X ⊕ Y

For the BORROW bit

B = not-X AND Y = X.Y

If we compare the Boolean expressions of the half subtractor with a half adder, we can see that the two expressions for the SUM (adder) and DIFFERENCE (subtractor) are exactly the same and so they should be because of the Exclusive-OR gate function. The two Boolean expressions for the binary subtractor BORROW is also very similar to that for the adders CARRY. Then all that is needed to convert a half adder to a half subtractor is the inversion of the minuend input X.

One major disadvantage of the Half Subtractor circuit when used as a binary subtractor, is that there is no provision for a “Borrow-in” from the previous circuit when subtracting multiple data bits from each other. Then we need to produce what is called a “full binary subtractor” circuit to take into account this borrow-in input from a previous circuit.

Answer 2

ANSWER OF 1;

Write down the truth table. The output is a 1 if the combination of inputs is odd, otherwise 0.

1. A B C ODD
2. 0 0 0 0
3. 0 0 1 1
4. 0 1 0 1
5. 0 1 1 0
6. 1 0 0 1
7. 1 0 1 0
8. 1 1 0 0
9. 1 1 1 1

Now at this point you could do a formal analysis and develop a minimised logic equation, but from the examination of the truth table, you can immediately observe the following:

• The first four outputs are the inverse of the second four, and that corresponds to the state of A
• The first four outputs are the XOR of B and C
• The second four outputs are XNOR of B and C

The XOR gate can be thought of as a controlled inverter - if one of its inputs is 0, the output follows the other input, if it’s 1, the output is the inverse of the input. So to implement this we just need two XOR gates:

ANSWER 2;

As their name implies, a Binary Subtractor is a decision making circuit that subtracts two binary numbers from each other, for example, X – Y to find the resulting difference between the two numbers.

Unlike the Binary Adder which produces a SUM and a CARRY bit when two binary numbers are added together, the binary subtractor produces a DIFFERENCE, D by using a BORROW bit, B from the previous column. Then obviously, the operation of subtraction is the opposite to that of addition.

We learnt from our maths lessons at school that the minus sign, “–” is used for a subtraction calculation, and when one number is subtracted from another, a borrow is required if the subtrahend is greater than the minuend. Consider the simple subtraction of the two denary (base 10) numbers below.

123	X	(Minuend)
– 78	Y	(Subtrahend)
45	DIFFERENCE

We can not directly subtract 8 from 3 in the first column as 8 is greater than 3, so we have to borrow a 10, the base number, from the next column and add it to the minuend to produce 13 minus 8. This “borrowed” 10 is then return back to the subtrahend of the next column once the difference is found. Simple school math’s, borrow a 10 if needed, find the difference and return the borrow.

The subtraction of one binary number from another is exactly the same idea as that for subtracting two decimal numbers but as the binary number system is a Base-2 numbering system which uses “0” and “1” as its two independent digits, large binary numbers which are to be subtracted from each other are therefore represented in terms of “0’s” and “1’s”.

Binary Subtraction

Binary Subtraction can take many forms but the rules for subtraction are the same whichever process you use. As binary notation only has two digits, subtracting a “0” from a “0” or a “1” leaves the result unchanged as 0-0 = 0 and 1-0 = 1. Subtracting a “1” from a “1” results in a “0”, but subtracting a “1” from a “0” requires a borrow. In other words 0 – 1 requires a borrow.

Binary Subtraction of Two Bits

0	1	1	(borrow)1→ 0
– 0	– 0	– 1	– 1
0	1	0	1

For the simple 1-bit subtraction problem above, if the borrow bit is ignored the result of their binary subtraction resembles that of an Exclusive-OR Gate. To prevent any confusion in this tutorial between a binary subtractor input labelled, B and the resulting borrow bit output from the binary subtractor also being labelled, B, we will label the two input bits as X for the minuend and Y for the subtrahend. Then the resulting truth table is the difference between the two input bits of a single binary subtractor is given as:

2-input Exclusive-OR Gate

Symbol	Truth Table
2-input Ex-OR Gate	Y	X	Q
	0	0	0
	0	1	1
	1	0	1
	1	1	0

As with the Binary Adder, the difference between the two digits is only a “1” when these two inputs are not equal as given by the Ex-OR expression. However, we need an additional output to produce the borrow bit when input X = 0 and Y = 1. Unfortunately there are no standard logic gates that will produce an output for this particular combination of X and Y inputs.

But we know that an AND Gate produces an output “1” when both of its inputs X and Y are “1” (HIGH) so if we use an inverter or NOT Gate to complement the input X before it is fed to the AND gate, we can produce the required borrow output when X = 0 and Y = 1 as shown below.

Then by combining the Exclusive-OR gate with the NOT-AND combination results in a simple digital binary subtractor circuit known commonly as the Half Subtractor as shown.

A Half Subtractor Circuit

A half subtractor is a logical circuit that performs a subtraction operation on two binary digits. The half subtractor produces a sum and a borrow bit for the next stage.

Half Subtractor with Borrow-out

Symbol	Truth Table
	Y	X	DIFFERENCE	BORROW
	0	0	0	0
	0	1	1	0
	1	0	1	1
	1	1	0	0

From the truth table of the half subtractor we can see that the DIFFERENCE (D) output is the result of the Exclusive-OR gate and the Borrow-out (Bout) is the result of the NOT-AND combination. Then the Boolean expression for a half subtractor is as follows.

For the DIFFERENCE bit:

D = X XOR Y = X ⊕ Y

For the BORROW bit

B = not-X AND Y = X.Y

If we compare the Boolean expressions of the half subtractor with a half adder, we can see that the two expressions for the SUM (adder) and DIFFERENCE (subtractor) are exactly the same and so they should be because of the Exclusive-OR gate function. The two Boolean expressions for the binary subtractor BORROW is also very similar to that for the adders CARRY. Then all that is needed to convert a half adder to a half subtractor is the inversion of the minuend input X.

One major disadvantage of the Half Subtractor circuit when used as a binary subtractor, is that there is no provision for a “Borrow-in” from the previous circuit when subtracting multiple data bits from each other. Then we need to produce what is called a “full binary subtractor” circuit to take into account this borrow-in input from a previous circuit.

Answer 3

ANSWER OF 1;

Write down the truth table. The output is a 1 if the combination of inputs is odd, otherwise 0.

1. A B C ODD
2. 0 0 0 0
3. 0 0 1 1
4. 0 1 0 1
5. 0 1 1 0
6. 1 0 0 1
7. 1 0 1 0
8. 1 1 0 0
9. 1 1 1 1

Now at this point you could do a formal analysis and develop a minimised logic equation, but from the examination of the truth table, you can immediately observe the following:

• The first four outputs are the inverse of the second four, and that corresponds to the state of A
• The first four outputs are the XOR of B and C
• The second four outputs are XNOR of B and C

The XOR gate can be thought of as a controlled inverter - if one of its inputs is 0, the output follows the other input, if it’s 1, the output is the inverse of the input. So to implement this we just need two XOR gates:

ANSWER 2;

As their name implies, a Binary Subtractor is a decision making circuit that subtracts two binary numbers from each other, for example, X – Y to find the resulting difference between the two numbers.

Unlike the Binary Adder which produces a SUM and a CARRY bit when two binary numbers are added together, the binary subtractor produces a DIFFERENCE, D by using a BORROW bit, B from the previous column. Then obviously, the operation of subtraction is the opposite to that of addition.

We learnt from our maths lessons at school that the minus sign, “–” is used for a subtraction calculation, and when one number is subtracted from another, a borrow is required if the subtrahend is greater than the minuend. Consider the simple subtraction of the two denary (base 10) numbers below.

123	X	(Minuend)
– 78	Y	(Subtrahend)
45	DIFFERENCE

We can not directly subtract 8 from 3 in the first column as 8 is greater than 3, so we have to borrow a 10, the base number, from the next column and add it to the minuend to produce 13 minus 8. This “borrowed” 10 is then return back to the subtrahend of the next column once the difference is found. Simple school math’s, borrow a 10 if needed, find the difference and return the borrow.

The subtraction of one binary number from another is exactly the same idea as that for subtracting two decimal numbers but as the binary number system is a Base-2 numbering system which uses “0” and “1” as its two independent digits, large binary numbers which are to be subtracted from each other are therefore represented in terms of “0’s” and “1’s”.

Binary Subtraction

Binary Subtraction can take many forms but the rules for subtraction are the same whichever process you use. As binary notation only has two digits, subtracting a “0” from a “0” or a “1” leaves the result unchanged as 0-0 = 0 and 1-0 = 1. Subtracting a “1” from a “1” results in a “0”, but subtracting a “1” from a “0” requires a borrow. In other words 0 – 1 requires a borrow.

Binary Subtraction of Two Bits

0	1	1	(borrow)1→ 0
– 0	– 0	– 1	– 1
0	1	0	1

For the simple 1-bit subtraction problem above, if the borrow bit is ignored the result of their binary subtraction resembles that of an Exclusive-OR Gate. To prevent any confusion in this tutorial between a binary subtractor input labelled, B and the resulting borrow bit output from the binary subtractor also being labelled, B, we will label the two input bits as X for the minuend and Y for the subtrahend. Then the resulting truth table is the difference between the two input bits of a single binary subtractor is given as:

2-input Exclusive-OR Gate

Symbol	Truth Table
2-input Ex-OR Gate	Y	X	Q
	0	0	0
	0	1	1
	1	0	1
	1	1	0

As with the Binary Adder, the difference between the two digits is only a “1” when these two inputs are not equal as given by the Ex-OR expression. However, we need an additional output to produce the borrow bit when input X = 0 and Y = 1. Unfortunately there are no standard logic gates that will produce an output for this particular combination of X and Y inputs.

But we know that an AND Gate produces an output “1” when both of its inputs X and Y are “1” (HIGH) so if we use an inverter or NOT Gate to complement the input X before it is fed to the AND gate, we can produce the required borrow output when X = 0 and Y = 1 as shown below.

Then by combining the Exclusive-OR gate with the NOT-AND combination results in a simple digital binary subtractor circuit known commonly as the Half Subtractor as shown.

A Half Subtractor Circuit

A half subtractor is a logical circuit that performs a subtraction operation on two binary digits. The half subtractor produces a sum and a borrow bit for the next stage.

Half Subtractor with Borrow-out

Symbol	Truth Table
	Y	X	DIFFERENCE	BORROW
	0	0	0	0
	0	1	1	0
	1	0	1	1
	1	1	0	0

From the truth table of the half subtractor we can see that the DIFFERENCE (D) output is the result of the Exclusive-OR gate and the Borrow-out (Bout) is the result of the NOT-AND combination. Then the Boolean expression for a half subtractor is as follows.

For the DIFFERENCE bit:

D = X XOR Y = X ⊕ Y

For the BORROW bit

B = not-X AND Y = X.Y

If we compare the Boolean expressions of the half subtractor with a half adder, we can see that the two expressions for the SUM (adder) and DIFFERENCE (subtractor) are exactly the same and so they should be because of the Exclusive-OR gate function. The two Boolean expressions for the binary subtractor BORROW is also very similar to that for the adders CARRY. Then all that is needed to convert a half adder to a half subtractor is the inversion of the minuend input X.

One major disadvantage of the Half Subtractor circuit when used as a binary subtractor, is that there is no provision for a “Borrow-in” from the previous circuit when subtracting multiple data bits from each other. Then we need to produce what is called a “full binary subtractor” circuit to take into account this borrow-in input from a previous circuit.